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A number when divided by 53 gives 34 as quotient and 21 as remainder. Find the number.

In Euclid's division lema a=bq+r, where O$\le r<b$. What is a?

The product of two consecutive positive integers is divisible by 2. Is this statement true or false? Give reason

Show that any positive integer is of the form 3q or 3q+1 or 3q+2 for some integer  q.

Write whether every positive integer can be of the form 4q+2, where q is an integer. Justify your answer.

Show that the square of an odd positive integer is of the form 8m+1, where m is some whole number.

Use the Euclid's division algorithm to find the HCF of
1. 650 and 1170
2. 870 and 225

Use Euclid's division algorithm to the HCF of the following three numbers.
1. 441, 567 and 693
2. 1620, 1725 and 255

Three pieces of timber 42 m, 49 m and 56 m long have to be divided into planks of the same length. What is the greatest possible length of each plank?
 

Write the missing numbers is the following factorisation.
1. 
2. 

See the following factor tree for factorisation of 156. Find the value of a.

Complete the following factor tree and find the composite number x.

Factorise the following number through the tree.
1. 420
2. 468
3. 308
4. 528

Express each of the following integers as a product of its prime factors
1. 945
2. 204
3. 660
4. 7325
5. 99792
6. 874944

Explain whether the following numbers are prime or composite numbers.
1. 7 x 11 x 13 + 13
2. (3 x 5 x 13 x 46)+23

Can the number ${16}^n$, n being a natural number, end with the digit 0? Give reason.
 

If two positive integers a and b are written as 
$\mathrm{a}={\mathrm{x}}^3{\mathrm{y}}^2$ and $\mathrm{b}={\mathrm{xy}}^3$; where. x and y are prime numbers, find the HCF of a and b.

If two positive integers p and q can be expressed as p=$a{b}^2$ and q=$a^{3}b$; where a, b being prime numbers, find the LCM (p, q).

Find the HCF of 96 and 404 by the prime factorisation method.
 

Factorise 612 ans 1314 by using tree method and find HCF and LCM

Find the HCF and LCM of 60, 84 and  108 by using the prime factorisation method.
 

Find the greatest possible length which can be used to measure exactly the length 7m, 3m 85 cm and 12 m 95 cm.

If the LCM of 26 and 91 and 182, find their HCF.

If the HCF of 150 and 100 is 50, find the LCM of 150 and 100.

The HCF of two numbers is 113 and their LCM is 56952. If one number is 904, find the other number

If HCF (253, 440)=11 and
LCM (253, 440)=253 x R. Find the value of R.

Find the HCF of 26 and 455. With the help of HCF, Find LCM also.

If LCM of 12 and 42 is 10m+4, Find the value of m.

If HCF of 210 and 55 is expressible in the form 210 x 5-55x, find the value of x.

Find the least number that is divisible by all the numbers from 1 to 10  (both inclusive).

Is product of a rational number and an irrational number, a rational number? Is product of two irrational number, rational numbers or irrational number? Justify by giving examples

If $\frac{7}{625}$ is a rational number, find the decimal expansion of it, which terminate.

Express the number $0.3\overline{178}$ in the form of rational number a/b.

Without actually performing the long division state whether $\frac{543}{225}$ has a terminating decimal expansions or non-terminating recurring decimal expansion.

The decimal expansion of the rational number $\frac{43}{2^4\times 5^3}$ will terminate after how many places of decimal?

Write the denominator of the rational number 257/500 in the form $2^m\times 5^n$, where m and n are non-negative integers. Hence write its decimal expansion without actual division.

If $\frac{241}{4000}=\frac{241}{2^m{5}^n}$, then find the values of m and n, where m and n are non-negative integers. Hence, write its decimal expansion without actual division.

Express $\left(\frac{15}{4}+\frac{5}{40}\right)$ as a decimal without actual division.

What can you say about the prime factorisation of the denominators?
1. 34.12345
2. $34.\overline{5678}$

Use Euclid's division algorithm to find the HCF of
1. 135 and 225
2. 196 and 38220
3. 867 and 255

Show that any positive odd integer is of the form 6q+1 or 6q+3 or 6q+5 where q is some integer.

An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of coloumns in which they can march?

Use Euclid's division lemma to show that the square of any positive integer is either of the form 3m or 3m+1, for some integer m.

Use Euclid's division lemma to show that the cube of any positive integer is either of the form 9m or 9m+1 or 9m+8

Express each number as a product of its prime fators
1. 140
2. 156
3. 3815
4. 5005
5. 7429

Find the LCM and HCF of the following pairs of integers and verify that
LCM x HCF=Product of two numbers.
1. 26 and 91
2. 510 and 92
3. 336 and 54

Find the HCF and LCM of the following integers by applying the prime factorisation method. 
1. 12, 15 and 21
2. 17, 23 and 29
3. 8, 9 and 25

If HCF of 306 anf 657 is 9, then find their LCM.

Check whether $6^n$ can end with the digit 0 for any natural number n.
 

Explain, why (7 x 11 x 13)+13 and (7 x 6 x 5 x 4 x 3 x 2 x 1)+5 are composite numbers?

There is a circular path around a sports field. Sonia takes 18 min to drive one round of the field , while Ravi takes 12 min for the same. Suppose they both start at the same point and at the same time and go in the same direction. After how many minutes will they meet again at the starting point?

Prove that $\sqrt{5}$ is an irrational number.
 

Prove that $3+2\sqrt{5}$ is irrational.

Prove that the following are irrational
1. $\frac{1}{\sqrt{2}}$
2. $7\sqrt{5}$

Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion.
1. $\frac{13}{3125}$
2. $\frac{17}{8}$
3. $\frac{64}{455}$
4. $\frac{15}{1600}$
5. $\frac{29}{343}$
6. $\frac{23}{2^3{5}^2}$
7. $\frac{129}{2^2{5}^7{7}^5}$
8. $\frac{6}{15}$
9. $\frac{35}{50}$
10. $\frac{77}{210}$
 

Write down the decimal expansions of those rational numbers in 
1. $\frac{13}{3125}$
2. $\frac{17}{8}$
3. $\frac{15}{1600}$
4. $\frac{23}{2^3{5}^2}$
5. $\frac{6}{15}$
6. $\frac{35}{50}$
Which have terminating decimal expansions.

The following reals numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational and of the form p/q. then what we can say about the prime factor of q?
1. 43. 123456789
2. 0.120120012000120000....
3. $43.\overline{123456789}$

By what number should 1365 be divided to get 31 quotient and 32 as  remainder?

The product of three consecutive positive integers is divisible by6. Is this statement true or false? Justify your answer.

If the HCF of 65 and 117 is expressible in the form 65m-117, then find the value of m.

Find the largest number which divides 70 and 125 leaving remainder 5 and 8 respectively.

Express $0.5\overline{4}$ recurring decimals as fraction in their lowest term.

In Euclid's division lemma, the value of r, when a positive integer a is divided by 3, are 0 and 1only. Is the statement true or false? Justify your answer.

On GT road, three consecutive traffic lights change after 36, 42 and 72 s. If the lights are first switched on at 9:00 am, then at what time will they change simultaneously?

For what value of n, $2^n$ x $5^n$ ends with 5?
 

Can the number $6^n$, where n being a natural number, ends with digit 5? Give reason.

Write the HCF and LCM of the smallest odd composite number and the smallest odd prime number. If an odd number p divides $q^2$, then will it divide $q^3$ also? Explain.

A rational number in its decimal expansion is 327.7081. What can you say about the prime factors of q, when this number is expressed in the  form $\frac{p}{q}$? Give reason.

If n is an odd integer, then show that $n^2-1$ is divisible by 8.

Without actually performing the long division, find if $\frac{987}{10500}$ will have terminating or non-terminating (repeating) decimal expansion. Give reason for your answer.

A forester wants to plant 66 mango trees, 88 orange trees and 110 apple trees in equal rows (in terms of number of trees). Also, he wants to make distinct rows of trees (i.e. only one type of trees in one row). Find the number of minimum rows.

Write whether the square of any positive integer can be of the form 3m+2, where m is a natural number. Justify your answer.

The numbers 525 and 3000 are both divisile by 3, 5 15, 25 and 75. What is the HCF of 525 and 3000? Justify your answer.

Express $5.4\overline{178}$ in the form $\frac{p}{q}$.

Find the greatest number which on dividing 1657 and 2037 leaves remainders 6 and 5 respectively.

State whether $1.2\overline{3}$$+\frac{3}{4}$ is a rational number or not.

If q is prime than prove that $\sqrt{q}$ is an irrational number.

A trader was moving along a road selling eggs. An idler who did not have much work to do, started to get the trader into a world duel. This grew into a fight, he pulled the basket with eggs and dashed it on the floor. The eggs broke. The trader requested the panchayat to ask the idler to pay for the broken eggs. The panchayat asked the trader how many eggs were broken?
He gave the following response 
If counted in pairs one will remain. If counted in 3's two will remain. If counted in 5's, four will remain. If counted in 6's five will remain. If counted in 7's nothing will remain and my basket will not accomodate more than 150 eggs.
1. How many eggs were there?
2. Which mathematical concept is used to solve the above problem?
3. What is the value shown by the trader in the question?